openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /LinearAlgebra /FunctionalCoercive.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace WeakMTWTransport | |
| section FunctionalCoercive | |
| variable {E:Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| lemma functional_rank_one_positive | |
| {L V:E →L[ℝ] E →L[ℝ] ℝ} {f g:E →L[ℝ] ℝ} {e:E} {β m γ:ℝ} | |
| (hLs:∀d e,L d e=L e d) (hL:∀d,0 ≤ L d d) | |
| (hV:∀d,d≠0 → 0 < V d d) | |
| (hVL:∀d,V d d ≤ L d d+β*(f d)^2) | |
| (hker:∀d,L e d=0) (hf:f e≠0) (hg:g e≠0) | |
| (hm:0 < m) (hγ:0 < γ) : | |
| ∀d:E,d≠0 → 0 < m*L d d+γ*(g d)^2 := by | |
| intro d hd | |
| have hnon:0 ≤ m*L d d+γ*(g d)^2:= | |
| add_nonneg (mul_nonneg hm.le (hL d)) (mul_nonneg hγ.le (sq_nonneg _)) | |
| by_contra H | |
| have hz:m*L d d+γ*(g d)^2=0:=le_antisymm (not_lt.mp H) hnon | |
| have hLd:L d d=0:=by | |
| have :m*L d d=0:=by nlinarith only [hz,mul_nonneg hm.le (hL d),mul_nonneg hγ.le (sq_nonneg (g d))] | |
| exact (mul_eq_zero.mp this).resolve_left hm.ne' | |
| have hgd:g d=0:=by | |
| have :γ*(g d)^2=0:=by rw [hLd,mul_zero,zero_add] at hz; exact hz | |
| exact sq_eq_zero_iff.mp ((mul_eq_zero.mp this).resolve_left hγ.ne') | |
| let a:=f d/f e | |
| let w:=d-a • e | |
| have hfw:f w=0:=by | |
| dsimp only [w,a] | |
| rw [map_sub,map_smul,smul_eq_mul,div_mul_cancel₀ _ hf,sub_self] | |
| have hLw:L w w=0:=by | |
| dsimp only [w] | |
| simp only [map_sub,map_smul,sub_apply,smul_apply, | |
| smul_eq_mul,hker,hLs d e,hLd,mul_zero,sub_zero] | |
| have hw:w=0:=by | |
| by_contra hw | |
| have H:=hVL w | |
| rw [hfw,hLw,pow_two,mul_zero,mul_zero,add_zero] at H | |
| exact (not_lt_of_ge H) (hV w hw) | |
| have hdw:d=a • e:=sub_eq_zero.mp hw | |
| have ha:a=0:=by | |
| rw [hdw,map_smul,smul_eq_mul] at hgd | |
| exact (mul_eq_zero.mp hgd).resolve_right hg | |
| exact hd (by rw [hdw,ha,zero_smul]) | |
| end FunctionalCoercive | |
| end WeakMTWTransport | |
| end | |
| end | |
| end OAI | |